Track / Stage 6 / 306 of 400 #1306 of 1964

Problem 1306

National Olympiad, first round

Algebra Difficulty 6.4 Prove it

Let $x_{1}, x_{2}, \cdots, x_{n}$ be positive numbers, and $\sum_{i=1}^{n} x_{i}=1$ , prove that:
$\left(\sum_{i=1}^{n} \sqrt{x_{i}}\right)\left(\sum_{i=1}^{n} \frac{1}{\sqrt{1+x_{i}}}\right) \leqslant \frac{n^{2}}{\sqrt{n+1}}$

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

None

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.